THE DOUBTEB less than 2 "with coefficients of the prescribed kind; in fact 1 _- V - 5 is the root of a; - (1 -f V~o) = 0, and the last coefficient, — (1 -f V — 5), is not a rational integer. If in the above definition of an algebraic integer of degree » we suppress the requirement that the leading coefficient be 1, and say that it can be any rational integer (other than zero, which is considered an integer), a root of the equation is then called an algebraic number of degree n. Thus |(1 -f V — 5) is an algebraic number of degree 2, but is not an algebraic integer; it is a root of 2x* - 2# + 3 = 0. Another concept, that of an algebraic number field of degreen, is now introduced: if r is an algebraic number of degree n, the totality of all expressions that can be constructed from r by repeated additions, subtractions, multiplications, and divisions (division by zero is not defined and hence is not attempted or permitted), is called the algebraic number field generated by r, and may be denoted by Ffr], For example, from r we get r + r, or 2r; from this and r we get 2r/r or 2, 2r — r or r, 2r x r or 2r2, etc. The degree of this F[r] is n. It can be proved that every member of F[r\ is of the form Co*"""1 -r C3/n~2 4* - • - T- c«_i? where the c's are rational numbers, and further every member of F[r] is an algebraic number of degree not greater than n (in fact the degree is some divisor of n). Some, but not all, algebraic numbers hi JP[r] will be algebraic integers. The central problem of the theory of algebraic numbers is to investigate the laws of arithmetical divisibility of algebraic integers in an algebraic number field of degree n. To make this problem definite it is necessary to lay down exactly what is meant by 'arithmetical divisibility', and for this we must understand the like for the rational integers. "We say that one rational integer, m, is divisible by another, d, if we canfindarationalinteger,#, such that m — q X d;d (also g) is called a divisor of m. For example 6 is a divisor of 12, because 12 = 2 x 6; 5 is not a divisor of 12 because there does not exist a rational integer q such that 12 = q x 5, A (positive) rational prime is a rational integer greater than 1 MJL—TOL. n. H 519